Peak to Average Power Ratio and CCDF

Peak to Average Power Ratio (PAPR) is often used to characterize digitally modulated signals. One example application is setting the level of the signal in a digital modulator. Knowing PAPR allows setting the average power to a level that is just low enough to minimize clipping.

However, for a random signal, PAPR is a statistical quantity. We have to ask, what is the probability of a given peak power? Then we can decide where to set the average power level such that clipping occurs at an acceptably low rate.

Let Pratio = the ratio of the instantaneous power of a sample to the average power of the signal. The probability that Pratio exceeds a given value is called the Complementary Cumulative Distribution Function (CCDF). Given a signal x, the CCDF y is computed as follows:

In the attached m-file, we calculate the CCDF for a sine wave and for Gaussian noise. Since the sine wave is periodic, we need only use a single period. For the noise, we need a lot of samples to find the low probability amplitudes: using 100,000 samples yields 7 samples with 10log10(Pratio) > 12 dB.

The figure below shows CCDF for a sine wave and for 100,000 samples of Gaussian noise. The PAPR of the sine wave is the maximum value vs. the x-axis. This is of course 20*log10[Vpeak/Vrms] = 20*log10[sqrt(2)] = 3.01 dB.